Arithmetic or Geometric? a) If a_{1}, a_{2}, a_{3}, \cdots is an

usagirl007A 2021-08-13 Answered
Arithmetic or Geometric?
a) If a1,a2,a3, is an arithmetic sequence, is the sequence a1+2,a2+2,a3+2, arithmetic?
b) If a1,a2,a3, is a geometric sequence, is the sequence 5a1,5a2,5a3, geometric?
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Khribechy
Answered 2021-08-14 Author has 100 answers
a) To show: the sequence a1+2,a2+2,a3+2, is arithmetic.
Given:
The sequence a1,a2,a3, is an arithmetic sequence.
Approach:
In arithmetic sequence, difference between the terms remains constant throughout.
Calculation:
The sequence a1,a2,a3, is an arithmetic sequence.
a2a1=a3a2=(1)
(a2+2)(a1+2)=(a2a1)
(a3+2)(a2+2)=(a3a2)
From equation (1)
(a3+2)(a2+2)=(a2a1)
As the differences between the terms are same then it is an arithmetic sequence.
Conclusion: hence, the sequence a1+2,a2+2,a3+2, is arithmetic.
b) To show: 5a1,5a2,5a3, is a geometric sequence.
Given: The sequence a1,a2,a3, is geometric sequence.
Approach: In a geometric sequence each term is found by multiplying the previous term by a constant.
Calculation:
a2a1=a3a2=(1)
Multiplying equation (1) by 5
5a25a1=5a35a2=
Then it is a geometric sequence.
Conclusion: hence, the sequence 5a1,5a2,5a3, is a geometric sequence.
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